Finite-rank Bratteli-Vershik diagrams are expansive—a new proof

  • Siri-Malén Høynes

Abstract

Downarowicz and Maass (Ergod. Th. and Dynam. Sys. 28 (2008), 739–747) proved that the Cantor minimal system associated to a properly ordered Bratteli diagram of finite rank is either an odometer system or an expansive system. We give a new proof of this truly remarkable result which we think is more transparent and easier to understand. We also address the question (Question 1) raised by Downarowicz and Maass and we find a better (i.e. lower) bound. In fact, we conjecture that the bound we have found is optimal.

References

Downarowicz, T. and Maass, A., Finite-rank Bratteli-Vershik diagrams are expansive, Ergodic Theory Dynam. Systems 28 (2008), no. 3, 739–747. https://doi.org/10.1017/S0143385707000673

Giordano, T., Putnam, I. F., and Skau, C. F., Topological orbit equivalence and $C^*$-crossed products, J. Reine Angew. Math. 469 (1995), 51–111.

Herman, R. H., Putnam, I. F., and Skau, C. F., Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3 (1992), no. 6, 827–864. https://doi.org/10.1142/S0129167X92000382

Hewitt, E. and Ross, K. A., Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematischen Wissenschaften, Bd. 115, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963.

Høynes, S.-M., Toeplitz flows and their ordered K-theory, Ergodic Theory Dynam. Systems 36 (2016), no. 6, 1892–1921. https://doi.org/10.1017/etds.2014.144

Walters, P., An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982.

Published
2017-05-27
How to Cite
Høynes, S.-M. (2017). Finite-rank Bratteli-Vershik diagrams are expansive—a new proof. MATHEMATICA SCANDINAVICA, 120(2), 195-210. https://doi.org/10.7146/math.scand.a-25613
Section
Articles